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Journal of the Physical Society of Japan

Monte Carlo Simulation for the In-Plane Susceptibility of 1D Spin 1/2 and 1XYModel

Toshifumi Sakaguchi, Kenn Kubo, Satoshi Takada

Published Mar 15, 1985• 24 citations
Publisher

Abstract

Monte Carlo calculation was performed for the in-plane susceptibility χ of 1D XY model with S =1/2 and S =1. The critical index γ of χ estimated from the result for S =1/2 agrees well with that (γ=1.5) derived from the exact spin-correlation given by McCoy. The critical index γ for S =1 is estimated to be very close to that (γ=2) of the classical ( S =∞) model.

DOI
10.1143/jpsj.54.861

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Monte Carlo calculation was performed for the in-plane susceptibility χ of 1D XY model with S =1/2 and S =1. The critical index γ of χ estimated from the result for S =1/2 agrees well with that (γ=1.5) derived from the exact spin-correlation given by McCoy. The critical index γ for S =1 is estimated to be very close to that (γ=2) of the classical ( S =∞) model.
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We study the classical XY (plane rotator) model at the Kosterlitz–Thouless phase transition. We simulate the model using the single-cluster algorithm on square lattices of a linear size up to L = 2048. We derive the finite-size behaviour of the second moment correlation length over the lattice size ξ 2nd / L at the transition temperature. This new prediction and the analogous one for the helicity modulus ϒ are confronted with our Monte Carlo data. This way β KT = 1.1199 is confirmed as inverse transition temperature. Finally, we address the puzzle of logarithmic corrections of the magnetic susceptibility χ at the transition temperature.
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Thermodynamic properties of the spin-1/2 Heisenberg ferromagnet are calculated by using Handscomb's Monte Carlo method. Several methods of analyses are used to determine critical properties of the model. For a square lattice we find that the susceptibility diverges exponentially at zero temperature. That is, \ensuremath{\chi}\ensuremath{\sim}exp[b(J/kB{\mathit{k}}_{\mathit{B}}kB​T)] with the constant b=4.5\ifmmode\pm\else\textpm\fi{}0.5, which is lower than the prediction (b=2\ensuremath{\pi}) of a modified spin-wave theory. For the simple-cubic lattice, we find that the critical temperature kB{\mathit{k}}_{\mathit{B}}kB​Tc{\mathit{T}}_{\mathit{c}}Tc​/J=1.68\ifmmode\pm\else\textpm\fi{}0.01, and the ratio of the exponents \ensuremath{\gamma}/\ensuremath{\nu}=2.0\ifmmode\pm\else\textpm\fi{}0.05, which are in good agreement with the estimates of the high-temperature series-expansion method.
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